﻿and Critical Speeds of Rotors. 153 



There are thus two natural frequencies of whirling, 

 depending on the direction in which whirling takes place. 

 For example, if the machine is running at what we shall 



presently see is the critical speed, namely, mo) = c 2 — \/ K , 



there are two possible frequencies of whirling for the free 

 vibration, viz. : g=l*618 C 2} or 0*618 C 2 . 

 8. The forced vibration is given by 



* ™V B 



» (J ,B- A) -Btf. . «****'•' * • (57) 



poa 



with a similar equation for r; in terms of sinpcot. 

 The amplitude of the vibration is a maximum when 



po> 2 (pB-A)-Bc 2 2 = 0, 

 or 



E) 2 fe 2 = / V A^ = I 1 M • < ' ( 58 ) 



' 2 p(pB — A) p(p-m)' 



9. This enables the second critical speed to be calculated 

 without difficulty, and fig. 6 gives the necessary curves for 

 reading off the proportional values directly. 



The ordinates give the values of co/c 2 and the abscissae the 

 values of the ratio A/B. The method of using the curves is 

 as follows : — 



1. Work out the radius of gyration ki about the shaft centre 



line. 



2. Work out the radius of gyration k 2 about a line perpen- 



dicular to this through the centre of gravity. 



3. Work out the ratio of A/B or k 1 /k 2 ; for turbo-generator 



rotors its value is usually between *2 and *4 and for 

 flywheels up to 2'0. (This gives the working point on 

 the horizontal axis of the curve.) 



4. Work out the *' first " critical speed in the usual way and 



multiply by ljk 2 , so as to obtain the stationary "second " 

 critical (where Z = half the distance between bearing 

 centres). 

 •5. To obtain the second critical speeds read off from the 

 curves the figures given on the vertical axis and 

 multiply by the stationary critical found from (4). 



